1. Technical Field of the Invention
The invention relates generally to communication systems; and, more particularly, it relates to modulation of signals that are transmitted within such communication systems.
2. Description of Related Art
Data communication systems have been under continual development for many years. A design imperative, particularly in the more recent days of broadband communications, is to maximize the amount of data (or information) that can reliably be communicated within a given period of time over a communication channel between a first point and a second point, e.g., the locations of a communication transmitter and a communication receiver.
It is well known since Shannon's original work in the late 1940's and subsequent interpretations that the maximal data rate supported by a communication channel increases linearly with the channel bandwidth and logarithmically with the SNR (signal-to-noise ratio) of the received signals. More precisely, the supported data rate depends linearly on the modulation rate, which is limited by the available channel bandwidth. Further, the supported data rate depends logarithmically on the SNR reduced by the so-called “SNR gap to capacity,” which depends on the sophistication of information encoding into signal waveforms at communication transmitters and decoding approaches employed in communication receivers.
A very significant portion of communication research has been devoted to reducing the SNR gap to capacity. For uncoded modulation, the gap to capacity is about 9 dB at moderately low error rate. Hence, due to the lack of sophisticated coding, for reliable information transmission the SNR must be 9 dB (decibels) higher than needed in an ideal system with capacity achieving modulation and coding in the sense of Shannon's theory. During the past two decades, this gap has been reduced to about 3 dB to 5 dB by the introduction of coded modulation, e.g., trellis coded modulation, and the extensive use of FEC (forward error correction) coding, e.g., Reed Solomon coding. More recently, “capacity approaching” coding techniques like turbo coding and decoding provide the possibility of reducing the SNR gap to capacity to less than 1.5 dB.
In comparison to work in coding research, efforts to achieve the highest possible modulation rate within given channel bandwidth have been of a more technical nature. As the modulation rate is increased towards the bandwidth of the communication channel, undesired ISI (inter-symbol interference) among serially transmitted modulation symbols is becoming harder to avoid. Similarly, suppression of interference from adjacent channels (ACI=adjacent channel interference) becomes more difficult.
Many of the presently deployed communication systems are based on the concept of zero-ISI transmission. In these systems, the modulation rate is 1/T=W/(1+α), where W is the channel bandwidth and α>0 is the fraction of “excess bandwidth” sacrificed to allow for smooth roll-offs of filter transfer functions towards zero outside of the transmission band. The quantity α is therefore also referred to as “spectral roll-off factor.” Typical values of α are in the range of 0.1 to 0.3. For given target values of ISI and ACI suppression, the complexity of transmit and receive filter increases with decreasing value of α. A chosen value of α reflects a compromise between filter complexity and efficient use of the available bandwidth.
FIG. 1 illustrates a prior art communication system operating at a modulation rate of 1/T=W/(1+α), across a communication channel of bandwidth W. In order to achieve zero ISI, the spectrum of the received signal must satisfy a condition known as the Nyquist criterion. Let H(f) be the Fourier transform of the received waveform in response to sending a single modulation symbol at time zero. This symbol response is considered at the output of the overall channel comprising transmit filter(s), communication channel, and receive filter(s). With constant delays ignored, the Nyquist criterion requires that H(f) and shifted versions of H(f) add up to a constant value, e.g.,
            ∑              k        =                  -          ∞                    ∞        ⁢                  ⁢          H      ⁡              (                  f          -                      k            T                          )              =      constant    .  To satisfy the criterion, H(f) must roll off in the symmetric fashion indicated in FIG. 1 around the edges of the 1/T band, which is called the Nyquist band. If the factor α were reduced to zero, such that 1/T=W, the spectral symbol response H(f) would have to assume rectangular shape. The required “brick wall” filtering cannot be realized with finite filter complexity.
In communication systems that operate using multiple bands, suppressing interference from signals transmitted in adjacent bands, in addition to suppressing out-of-band noise, is of major concern. FIG. 2 depicts the partitioning of a total bandwidth nW into n subbands of bandwidth W. Each subband is used for zero-ISI transmission at modulation rate 1/T.
FIG. 3 illustrates a prior art, complex baseband model of a zero-ISI baseline system with SQRC (square root raised-cosine) transmit and receive filters. An ideal communication channel with flat transfer function GC(f)=1 and constant PSD (power spectral density) N(f)=N0 (AWGN=additive white Gaussian noise) within the transmission band of the channel is assumed. The transfer functions of the transmit filter, HT(f), and the receive filter, GR(f), are given by HT(f)=GR(f)=√{square root over (H(f))}, where H(f) is the spectral RC (raised cosine) symbol response
                              H          ⁡                      (            f            )                          =                  T          ⁢                      {                                                                                                                                                        1                          ,                                                                                                                  f                                                                                      ≤                                                                                          1                                -                                α                                                                                            2                                ⁢                                                                                                                                  ⁢                                T                                                                                                              ,                                                                                                                                                                                                                        cos                              2                                                        ⁡                                                          (                                                                                                                                    π                                    ⁢                                                                                                                                                  ⁢                                    T                                                                                                        2                                    ⁢                                                                                                                                                  ⁢                                    α                                                                                                  ⁢                                                                  (                                                                                                                                                  f                                                                                                              -                                                                                                                  1                                        -                                        α                                                                                                                    2                                        ⁢                                        T                                                                                                                                              )                                                                                            )                                                                                ,                                                                                                                    1                                -                                α                                                                                            2                                ⁢                                                                                                                                  ⁢                                T                                                                                      ≤                                                                                        f                                                                                      ≤                                                                                          1                                +                                α                                                                                            2                                ⁢                                                                                                                                  ⁢                                T                                                                                                                                                                                                                                                                          0                    ,                    otherwise                                                                                                          EQ        .                                  ⁢                  (          1          )                    of the overall channel. One can easily verify that H(f) satisfies the Nyquist criterion, and ∫|HT(f)|2df=∫|GR(f)|2df=∫H(f)df=1.
In this system, complex-valued modulation symbols an are transmitted at a modulation rate of 1/T=W/(1+α). The modulation symbols are independently and identically distributed (i.i.d.) random variables with zero mean and average energy Ea=E{|an|}2). Sampling the output signal of the receive filter at times nT yields the ISI-free sequence of sample values yn=an+wn. The noise samples wn are complex-valued i.i.d. random variables with zero mean and variance σw2=E{|wn|2}=N0. The signal-to-noise ratio becomes SNR=Ea/N0. The baseline system then supports a data rate R of
                    R        =                              1            T                    ⁢                                    log              2                        ⁡                          (                              1                +                                  SNR                  G                                            )                                ⁢                                          ⁢          bit          ⁢                      /                    ⁢                      sec            .                                              EQ        .                                  ⁢                  (          2          )                    
In EQ. (2), G denotes the SNR gap from capacity. The value of G provides an abstract characterization of the effectiveness of particular types of modulation and coding schemes, with G=1 being the theoretical optimum. Assume real systems operating at BER (bit-error rate) of 10−6. Then, for uncoded modulation G≈8 (9 dB). Systems with currently widely employed modulation and coding are characterized by G≈4 to 2 (6 to 3 dB). With capacity approaching coding the SNR gap may eventually be reduced 1.5 dB or less.
Sequences of T-spaced quantities ζn can be written in polynomial sequence notation as ζ(D)=ΣnζnDn. The delay operator D may be expressed in the frequency domain by D=e−j2πfT (=z−1). Using this notation, the sequence of modulation symbols becomes a(D) and the sequence of received sample values can generally be written asy(D)=h(D)a(D)+w(D),  EQ. (3)where h(D) represents the symbol response. In the case of a zero-ISI system, h(D)=1 such that y(D)=a(D)+w(D).
In light of the above discussion, attempting to increase the modulation rate by reducing the amount of excess bandwidth inherently increases filter complexity as filters must approach a brick-wall characteristic to maintain zero ISI. Clearly, the realization of filters exhibiting such brick-wall characteristics may not be pursued. Therefore, there exists a need in the art to provide a solution that could enable the use of a modulation rate 1/T=W in a real communication system without necessitating the prohibiting operational constraint described above that requires zero-ISI transmission.